VI PKEFACE. equal to 1 4- 1/2+ 1/3+ . . . + l/(p — 1) is divisible by p2, a result first proved by Wolstenholme in 1862. Sylvester stated in 1866 that the sum of all products of n distinct numbers chosen from 1, 2, . . ., m is divisible by each prime >n+l which is contained in any term of the set m—n+1,. . .,m,m+l. There are various theorems analogous to these. In Chapter IV are given properties of the quotient (u^-ty/p, which plays an important r61e in recent investigations on Fermat's last theorem (the impossibility of xp+yps=zp if p>2), the history of which will be treated in the final chapter of Volume II. Some of the present papers relate to (tt*(»)— l)/n, where n is not necessarily a prime. While Euler's ^-function was defined above in order to state his generalization of Fermat's theorem, its numerous properties and generalizations are reserved for the long Chapter V. In 1801 Gauss gave the result that 0(<y + "• +<K&) = n, if d^ . . . , dk are the divisors of n; this was generalized by Laguerre hi 1872, H. G. Cantor hi 1880, Busche in 1888, Zsigmondy in 1893, Vahlen in 1895, Elliott in 1901, and Hammond hi 1916. In 1808 Legendre proved a simple formula for the number of integers ^ n which are divisible by no one of any given set of primes. The asymptotic value of 0(1) + . . . +0(6) for G large was discussed by Dirichlet in 1849, Mertens in 1874, Perott in 1881, Sylvester in 1883 and 1897, Cesaro in 1883 and 1886-8, Berger in 1891, and Kronecker in 1901. The solution of 4>(x) =?g was treated by Cayley in 1857, Minin in 1897, Pichler hi 1900, Carmichael in 1907-9, Ranum in 1908, and Cunningham in 1915. H. J. S. Smith proved in 1875 that the w-rowed determinant, having as the element in the ith row and jth column any function f(5) of the greatest common divisor 8 of i and j, equals the product of F(l)f F(2), . . ., F(m), where In particular, F(m)=<t>(m) if /(5)=5. In several papers (pp. 128-130) Cesaro considered analogous determinants. The fact that 30 is the largest number such that all smaller numbers relatively prime to it are primes was first proved by Schatunowsky in 1893. A. Thacker in 1850 evaluated the sum 4>k(n) of the kth powers of the integers ^n which are prime to n. His formula has been expressed in various symbolic forms by Cesaro and generalized by Glaisher and Nielsen. Crelle had noted in 1845 that 0x(w) = Jn0 ( n) . In 1869 Schemmel considered the number of sets of n consecutive integers each < m and prime to m. In connection with linear congruence groups, Jordan evaluated the number of different sets of k positive integers ^n whose greatest common divisor is prime to n. This generalization of Euler's 0-function has properties as simple as the latter function and occurs in many papers under a variety of notations. It in turn has been generalized (pp. 151-4).